What does the parameter $y$ stand for in function $g(y,\mu,\sigma)$ related to REINFORCE algorithm?
If you take a look at the Wikipedia page related to the normal distribution, you will see the definition of the Gaussian density
$$
{\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}} \label{1}\tag{1}
$$
and you will see that the $y$ in your formula corresponds to the $x$ in equation \ref{1}.
I've seen this notation in the context of computer vision and image processing, where the Gaussian kernel is used to blur images.
So, as pointed out by someone in a comment, $y$ should indeed be the point where you evaluate the density.
Maybe the confusing part is that all parameters are treated equally in terms of their purpose, while $\mu$ and $\sigma$ are clearly the parameters that define the specific density, so they are not the inputs to the specific density.
After having read the relevant section of the paper, I now understand why you're confused. The author refers to $y$ as the output (not yet sure why: maybe it's the output of another unit that feeds this Gaussian unit?), but I think that this explanation still applies. The output of the Gaussian density $g$ is not $y$, but the density that corresponds to $y$. In fact, in appendix $B$ of the paper, the author says that $Y$ is the support of $g$ and $y$ is an element of $Y$.